Regularity of ideals and their radicals (Q752767)

Let I be a homogeneous ideal of \(K[X_ 1,...,X_ n]\), K a field. This paper considers the question of when reg(\(\sqrt{I})\leq reg(I)\). (Here reg(I), the regularity of I, is defined to be the smallest integer k for which \([H^ i_{(X_ 1,...,X_ n)}(I)]_ j=0\) for all i,j\(\in {\mathbb{Z}}\) such that \(i+j\geq k+1.)\) The question is of interest since upper bounds given for reg(I) have depended on I being a radical ideal. The above inequality is shown to hold in the following three cases: \(R\setminus \sqrt{I}\) is a Buchsbaum R-module of depth at least one, I is generated by monomials in \(X_ 1,...,X_ n\), or in the case \(K={\mathbb{C}}\) where \(\sqrt{I}\) defines a nonsingular curve in \({\mathbb{P}}^ 3\) with degree at least 20 and greater than or equal to its genus.